A reduction formula is one that enables us to solve an integral problem by reducing it to a problem of solving an easier integral problem, and then reducing that to the problem of solving an easier problem, and so on.
For example, if we let
Integration by parts allows us to simplify this to
which is our desired reduction formula. Note that we stop at
Similarly, if we let
then integration by parts lets us simplify this to
Using the trigonometric identity, , we can now write
Rearranging, we get
Note that we stop at or 2 if is odd or even respectively.
As in these two examples, integrating by parts when the integrand contains a power often results in a reduction formula.